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Vol. 5 (2000) Paper no. 3, pages 1–16.
Journal URL
http://www.math.washington.edu/~ejpecp/
Paper URL
http://www.math.washington.edu/~ejpecp/EjpVol5/paper3.abs.html
STOCHASTIC WEAK ATTRACTOR FOR A DISSIPATIVE
EULER EQUATION
Hakima Bessaih
U.S.T.H.B. Institut de Mathimatiques
B.P 32 El-Alia Alger 16111 Algeria.
and Dipartimento di matematica applicata U.Dini
Via Bonanno 25/B 56126 Pisa (Italy)
bessaih@cibs.sns.it
Abstract In this paper a non autonomous dynamical system is considered, a stochastic one
that is obtained from the dissipative Euler equation subject to a stochastic perturbation, an
additive noise. Absorbing sets have been defined as sets that depend on time and attract from
−∞. A stochastic weak attractor is constructed in phase space with respect to two metrics and
is compact in the lower one.
Keywords Dissipative Euler equation, random dynamical systems, attractors.
Classification 2000 MSC 35Q35 37L55 60H15
Submitted to EJP on July 13, 1999. Final version accepted on November 29, 1999.
1
Introduction
In the classical theory of attractors, we need to define a semigroup S(t) on a metric space (X, d)
which must be continuous on X (continuity ensures the invariance of the attracting set). We
also need to have a concept of dissipativity and compactness for S(t). Usually, the semigroup
property follows from uniqueness.
Here, we are concerned with a stochastic dynamical system which is obtained from a partial
differential equation perturbed by a random forcing term. It is dissipative but does not produce regularization; the dissipative Euler equation. An existence and uniqueness theorem is
established when the initial data are in W (defined below). Dissipation occurs also in W, but
the continuity of the dynamic is not satisfied. Moreover the compactness of the flow does not
occur with respect to the topology of W, but in a lower sence. In a preceding paper see [3]
and following the approach of [13], we have proved the existence of an attracting set using the
dynamic of the shift operator in the space of paths. Hence, the non-uniqueness problem (we
took only that u0 ∈ V , defined below) was bypassed and a continuous semigroup was obtained.
We have also introduced a notion of attractor relative to a pair (d, δ) of topologies where the
attractor constructed was d-bounded and δ-compact (recall that Euler equation do not produce
regularization), so that we obtain a weak attractor in path space. In this paper, we aim to
construct a stochastic weak attractor in the phase space W for (4.1), where uniqueness holds.
In this way, the dynamic is well defined and enjoys the evolution property. However, we are
not able to prove its continuity. So, the results of [3] concerning the attractor in the weak sense
can not be used here. Notice that in the classical theory of attractors, the continuity of S(t)
is necessary to prove the invariance property of the attractor. Instead of continuity we will
give another property which ensures the invariance of the absorbing set and the existence of a
weak attractor in phase space (for a deterministic dynamical system). Equation (4.1) is non
autonomous, but the general theory for non autonomous deterministic systems can not be used
here. In [4] and [5], attracting sets have been defined as sets that depend on time and attract
any orbits from −∞. Indeed, in this paper we will prove existence of weak random attractor at
time 0, which attracts bounded sets from −∞.
The paper is organised as follows, in section 2 we give the general theory of weak attractor
for deterministic non-autonomous dynamical systems, that we extend to stochastic dynamical
systems in section 3. In section 4 we apply it to the particular case of stochastic dissipative
Euler equation (4.1).
2
Deterministic weak attractor for non autonomous systems
If d denotes the distance on a metric space X, for each pair (A, B) of subsets of X and each
x ∈ X, we define
d(x, A) = inf d(x, y),
d(B, A) = inf d(x, A).
y∈A
x∈B
Let W be a Banach metric space. We denote by dW the metric on W. Let us define a mapping
S(t, s) on (W, dW ), −∞ < s ≤ t < ∞. Assume that another metric δ exists on W. We introduce
2
the following concept of continuity for the mapping S(t, s);
if xn is dW − bounded and δ − convergent to x in W
then S(t, s)xn is δ − convergent to S(t, s)x, for all s ≤ t.
(2.1)
We assume that the family S(t, s) satisfy the condition (2.1) and the following evolution property,
S(t, r)S(r, s)x = S(t, s)x for all s ≤ r ≤ t and for all x ∈ W,
(2.2)
Definition 2.1 Given t ∈ R, we say that B(t) ⊂ W is a dW - absorbing set at time t if
1. B(t) is dW -bounded,
2. ∀B0 dW -bounded, ∃s1 (B0 ) such that S(t, s)B0 ⊂ B(t), ∀s ≤ s1 (B0 ).
Definition 2.2 Given t ∈ R, we say that {S(t, s)}t≥s is dW /δ-uniformly compact at time t if
for all B ⊂ W dW -bounded, there exists s0 which may depend on B such that
[
S(t, s)B
s≤s0
is relatively compact in W with the topology δ.
Definition 2.3 The family {S(t, s)}t≥s is asymptotically dW /δ-compact if there exists a dW absorbing set at time t, which is δ-compact
Definition 2.4 (Weak omega-limit set) For any set B ⊂ W, we define the δ-weak omega-limit
T
S
δ
set of B at time t and write ω δ (B, t), as the set τ ≤t s≤τ S(t, s)B , where the closure is taken
in the δ-topology of W (this set can be empty). It is characterized as follows: x ∈ ω δ (B, t) if
and only if there exists a sequence xn ∈ B and a sequence sn → −∞ such that
δ(S(t, sn )xn , x) → 0
as
n → ∞.
Definition 2.5 (Global weak attractor) We say that A(t) ⊂ W is a dW /δ global weak attractor
at time t if it verifies the following properties:
(i) A(t) is not empty, it is dW -bounded and δ-compact,
(ii) A(τ ) is invariant by S(τ, s), i.e. S(τ, s)A(s) = A(τ ) for all τ ≥ s ≥ t,
(iii) for every dW -bounded set B ⊂ W, lims→−∞ δ(S(t, s)B, A(t)) = 0.
Theorem 2.6 Let {S(t, s)}t≥s be a family of mappings on a metric space (W, dW ). Let δ be
another metric on W, such that S(t, s) : W → W verifies (2.1) and the evolution property, for
all t ≥ s. Assume that at time t, there exists a dW -absorbing set B(t), that it is δ-compact.
Then, there exists a dW /δ global weak attractor A(t), and A(t) =
is taken over all dW -bounded sets B)
3
S
δ
B∈W
ω δ (B, t) . (the union
Proof. Here, we must prove (i), (ii) and (iii) of definition (2.5).
(i) Notice that if there exists a dW -absorbing set B(t) at time t, then the set ω δ (B, t) ⊂ B(t)
is non-empty, dW -bounded and δ-compact, as a consequence also A(t), being the union over
dW -bounded sets.
(ii) Let us first prove that S(τ, s)A(s) ⊂ A(τ ). Let us take y ∈ A(s). By the caracterization of A(s), there exists a dW -bounded set Bn ⊂ W, a sequence yn ∈ ω δ (Bn , t) and a sequence
sn → −∞ such that yn −→δn→∞ y. This yields that
[
∀n ∈ N, ∀s0 ≤ s, yn ∈
δ
S(s, s0 )Bn .
s0 ≤s0
Hence, ∀n ∈ N, ∀s0 ≤ s, ∃sk ≤ s0 , znk ∈ Bn , such that
S(s, sk )znk →δk→∞ yn .
By (refevolution) and (2.1), we have that
S(τ, s)S(s, sk )znk = S(τ, sk )znk →δk→∞ S(τ, s)yn .
We deduce that
S(τ, s)yn ∈
\ [
δ
S(τ, r)Bn ⊂
s0 ≤τ r≤s0
[
\ [
δ
S(τ, r)B .
B⊂W s0 ≤τ r≤s0
On the other hand
S(τ, s)yn −→δn→∞ S(τ, s)y,
which implies that S(τ, s)y ∈ A(τ ).
Now let us prove that S(τ, s)A(s) ⊃ A(τ ). Taken x ∈ A(τ ), there exist a dW -bounded set
Bn ⊂ W, a sequence yn ∈ ω δ (Bn , t) such that yn →δ x and there exist a sequence ynk ∈ Bn and
a sequence sk → −∞ such that
S(τ, sk )ynk →δ yn .
On the other hand, there exists a dW -absorbing set at time t, that is B(t), so also at time s ≥ t
(take S(s, t)B(t) = B(s)), which attracts all bounded sets of Bn of W, i.e.
S(s, sk )ynk →dW φn ∈ B(s).
The absorbing set being δ-compact, the mapping S(τ, s) is also δ-compact, hence there exists a
0
subsequence S(s, sk0 )ynk , such that
0
S(s, sk0 )ynk −→δ φn .
Thus, φn ∈ ω δ (Bn , s). By the evolution property (2.2) and the (2.1) we get
0
S(τ, sk0 )ynk →δ S(τ, s)φn .
Hence S(τ, s)φn = yn . Since B(s) is δ-compact, there exists a subsequence φnk →δ φ and
φ ∈ A(s) and finally S(τ, s)φ = x. Hence x ∈ S(τ, s)A(s).
4
(iii) We argue by contradiction. Assume that there exist > 0, a dW -bounded set B and
sequences sn → −∞ and xn ∈ B such that
δ(S(t, sn )xn , ω δ (B, t)) ≥ > 0, as sn → −∞.
Since there exists a dW -absorbing set at time t, B(t), there exists a sequence yn ∈ B(t), such
that
dW (S(t, sn )xn , yn ) → 0, as sn → −∞.
B(t) being a δ-compact set, there exists a subsequence ynk → y and
δ(S(t, snk )xnk , y) → 0.
This yields that y ∈ ω δ (B, t), which contradicts the hypothesis.
3
Stochastic weak attractor
Let (W, dW ) a complete metric space and (Ω, F, P) a probability space. let us define another
metric δ on W. We consider a family of mapping {S(t, s.ω)}t≥s,ω∈Ω : W → W, satisfying for
P-a.e.ω ∈ Ω the properties (2.1) and (2.2).
Given t ∈ R and ω ∈ Ω, we say that B(t, ω) is dW -bounded absorbing set at time t if for all
dW -bounded set B ⊂ W, there exists s0 (B), such that
S(t, s, ω)B ⊂ B(t, ω)), ∀s ≤ s0 .
We say that {S(t, s.ω)}t≥s,ω∈Ω is asymptotically dW /δ-compact if there exists a measurable set
Ω0 ⊂ Ω with measure one, such that for all t ∈ R, and all ω ∈ Ω0 , there exists a dW -absorbing
set B(t, ω), δ-compact.
Let us denote by B(W) the σ-algebra of the metric space W. Let us give the following theorem
(see [16])
Theorem 3.1 Let X1 , X2 two metric spaces such that X1 ⊂ X2 with continuous embedding.
Then
\
B(X2 ) X1 ⊂ B(X1 ).
Let us denote by H the separable metric space such that
W ⊂ H,
with continuous embedding. The metric δ is the metric endowed on H and W is closed in H.
Assume the following condition
( M ) for all t ∈ R, x ∈ W, the mapping (s, ω) → S(t, s, ω)x is measurable from
T
(−∞, t] × Ω, B((−∞, t]) × F → (W, B(H) W).
5
As in section 2, we define the random weak omega-limit of a bounded set B ⊂ W at time t as:
ω δ (B, t, ω) =
\ [
S(t, s, ω)B
δ
τ <t s<τ
and
A(t, ω) =
[
δ
ω δ (B, t, ω) .
B⊂W
The set A(t, ω) will be called the random attractor. As in section 2, we give the following
theorem
Theorem 3.2 Assume the condition (M) holds, assume that for each t ∈ R there exists a measurable set Ωt ⊂ Ω with measure one, such that for all ω ∈ Ωt there exists an absorbing set,
δ-compact. Then for P.a.e. ω ∈ Ωt , the set A(t, ω) is a measurable global weak attractor, i.e.
(i) A(t, ω) is not empty, it is dW -bounded and δ-compact,
(ii) S(τ, s, ω)A(s, ω) = A(τ, ω) for all τ ≥ s ≥ t,
(iii) for every dW -bounded set B ⊂ W, lims→−∞ δ(S(t, s, ω)B, A(t, ω)) = 0.
(iv)A(t, ω) is measurable with respect to the P-completion of F.
Proof. (i), (ii) and (iii) are consequences of theorem of section 2.
(iv)We say that a family A(t, ω) of closed subsets of W is measurable if and only if for all x ∈ W,
the multifunction ω → δ(x, A(ω)) is measurable (see [6]).
For all x ∈ W, we have
δ(x, A(t, ω)) =
=
inf δ(x, ω δ (B, t, ω))
B⊂W
inf inf δ(x, S(t, s, ω)B).
B⊂W s≤t
The norm | · | being the norm endowed by H, (W, | · |) is a normed subspace of normed separable
space (H, | · |), hence (W, | · |) is also separable. By the assumption (M), for all t ∈ R and for
all yn in a dense subset of B ⊂ W, the function
(s, ω) → δ(x, S(t, s, ω)yn )
is measurable. By the property (2.1) and the separability of (W, | · |)
(s, ω) → δ(x, S(t, s)B)
is measurable. On the other hand, for each α ∈ R
ω ∈ Ω, inf δ(x, S(t, s)B) < α = ΠΩ {(s, ω) ∈] − ∞, t] × Ω, δ(x, S(t, s)B) < α} ,
s≤t
where ΠΩ is the canonical projection from R × Ω in Ω. We deduce by the projection theorem
(see [6]) that the set
ω ∈ Ω, inf δ(x, S(t, s)B) < α
s≤t
6
is measurable with respect to the P completion of F. By the separability of (W, |·|), we conclude
the measurability of the set
ω ∈ Ω, inf inf δ(x, S(t, s)B) < α .
B⊂W s≤t
This complete the proof.
4
4.1
Application to stochastic dissipative Euler equation.
Mathematical setting and notations
We are concerned with a stochastic dissipative Euler equation for an incompressible fluid in an
open bounded domain D of R2 , i.e.
du + ((u · ∇)u + χu)dt = (−∇p + f )dt + dW,
(4.1)
where u is the velocity of the fluid, p the pressure, f the external force, W is a Wiener process
on a complete probability space (Ω, F, P) (with expectation denoted by E). The constant χ will
be called the sticky viscosity. The term −χu does not correspond to a constitutive equation and
does not introduce the smoothing effects of the Navier-Stokes term ∆u. For other comments
see [10]. The velocity field u is in addition subject to the incompressibility condition
∇ · u(t, x) = 0,
t ∈ R,
x ∈ D,
(4.2)
the boundary condition
u · n = 0 on ∂D,
(4.3)
u(t0 ) = ut0 .
(4.4)
and it satisfies the initial condition
Let us introduce some functional spaces. Let V be the space of infinitely differentiable vector
fields u on D with compact support strictly contained in D, satisfying ∇ · u = 0. We introduce
the space H of all measurable vector fields u : D −→ R2 which are square integrable, divergence
free, and tangent to the boundary
h
i2
H = u ∈ L2 (D)
; ∇ · u = 0 in D, u · n = 0 on ∂D ;
the meaning of the condition u · n = 0 on ∂D for such vector fields is explained
for
instance in
2
2
[14]. The space H is a separable Hilbert space with the inner product of L (D) , denoted in
the sequel by < ., . > (norm |.|). Let V be the following subspace of H:
h
i2
V = u ∈ H 1 (D)
; ∇ · u = 0 in D, u · n = 0 on ∂D ;
2
The space V is a separable Hilbert space with the inner product of H 1 (D) (norm k . k).
Identifying H with its dual space H 0 , and H 0 with the corresponding natural subspace of the
dual space V 0 , we have the standard triple V ⊂ H ⊂ V 0 with continuous dense injections. We
7
denote the dual pairing between V and V 0 by the inner product of H.
In what follows, we will denote by d the metric endowed by V and by δ the metric endowed
by H. Let us denote by | · |p the norm endowed by the Lebesgue space Lp (D), when p 6= 2
From now on , we will assume that W (t) is an infinite dimensional Brownian motion of the form
P
W (t, ω) = ∞
j=1 σj βj (t, ω)ej where {βj } is a sequence of real independent towsided Brownian
motions on the probability space (Ω, F, P), and we will assume the the following regularity space
on W ; the process W = W (t, ω) ω ∈ Ω, is an H- valued process such that for P-a.e. ω ∈ Ω,
W (., ω) ∈ C R, [H 4 (D)]2
\
V
(4.5)
with the mapping (t, ω) → W (t, ω) measurable in these topologies, and
∇ ∧ W = 0 on R × ∂D,
(4.6)
where
∇ ∧ W = D1 W2 − D2 W1 .
The condition (4.5) can be written in this form
∞
X
λ4j σj2 < ∞,
(4.7)
j=1
where {λj }j and {ej }j are respectively the eigenvalues and the eigenfunctions of the linear
operator A defined below.
Let us introduce the unknown v = u − z (in spite of v = u − W , the reason is that we want to
use the ergodic properties of z), where z is solution of the following linear stochastic equation
dz + (χ + α)zdt = dW, α ≥ 0
(4.8)
so that v is solution of the deterministic equation
dv
+ (v + z) · ∇(v + z) + χv = −∇p + f + αz.
dt
4.2
(4.9)
Approximation scheme
Let us approximate (4.9) by the dissipative Navier-Stokes equations
 ∂v


∂t + (v + z) · ∇(v + z) + ∇p = ν∆v − χv + f + αz,



 ∇ · v = 0,
∇ ∧ v = 0,



v · n = 0,



v|t=t0 = ut0 ,
in (t0 , T ) × D
in (t0 , T ) × D
on (t0 , T ) × ∂D
on (t0 , T ) × ∂D
in D
with ν > 0. We have denoted by ∇ ∧ v the vorticity, defined as
∇ ∧ v = ∂2 v1 − ∂1 v2 .
Let us define the continuous bilinear form on V
Z
a(u, v) =
D
∇u · ∇v −
8
Z
∂D
k(σ)u(σ) · v(σ)dσ,
(4.10)
where k(σ) is the curvature. We set
n
o
D(A) = u ∈ V ∩ (H 2 (D))2 , ∇ ∧ u|∂D = 0 ,
and define the linear operator A : D(A) −→ H, as
< Au, v >= a(u, v), for all u, v inV.
Let us also define the trilinear form on V
Z
b(u, v, w) =
D
(u · ∇)v · w.
We define the bilinear operator B(u, v) : V × V −→ V 0 , as < B(u, v), z >= b(u, v.z) for all
z ∈ V . Note that B verifies < B(u, v), z >= − < B(u, z), v > and < B(u, v), v >= 0, when u,
v and z are in suitable spaces (see for instance [14]).
We will write (4.10) in the following abstract form
(
dvν
dt + νAvν +
vν (t0 ) = vt0 ,
B(vν + z, vν + z) + χvν = f + αz,
(4.11)
t ∈ [t0 , T ). Now, we consider the classical Faedo-Galerkin approximation scheme, we obtain the
following problem
(
dvnν
dt + νAvnν + Pn B(vnν
vnν (t0 ) = Pn vt0 .
+ zn , vnν + zn ) + χvnν = Pn f + αzn ,
(4.12)
where Pn is the orthogonal projector in H over the subspace spanned by e1 , e2 , ..., en , the first
n eigenfunctions of A, i.e.
Pn x =
n
X
< x, ei > ei , x ∈ H.
i=1
We can look for energy estimates satisfied by vnν . We multiply (4.12) by vnν and integrate over
D; by using the incompressibility condition,
|b(vnν + zn , vnν + zn , vnν )| = |b(vnν + zn , zn , vnν )|
≤ |
≤
Z
D
vnν · ∇zn · vnν | + |
Z
D
zn · ∇zn · vnν |
χ
|vnν |2 + C(χ)(|∇zn |2 |zn |2∞ + |vnν |2 |∇zn |2∞ ).
4
On the other hand there exists an arbitrary > 0, such that
Z
∂D
Therefore,
k(σ)u(σ) · v(σ)dσ ≤ k u k2 +C |v|2 .
|a(vnν , vnν )| ≤ |∇vnν |2 + |∇vnν |2 + C |vnν |2 .
Using the above estimates, we get
1d
−χ
|vnν (t)|2 + ν(1 − )|∇vnν (t)|2 ≤
|vnν (t)|2 + C |vnν (t)|2
2 dt
2
+ C(χ)(|f (t)|2 + |zn (t)|2∞ |∇zn (t)|2 + |vnν (t)|2 |∇zn (t)|2∞ ),
9
(4.13)
for an arbitrary > 0.
We integrate over (t0 , t). For all ν ≤ ν0 , and in particular for = 1/2, we obtain
2
|vnν (t)|
Z
t
+ ν
t0
2
|∇vnν (s)| ds ≤ |vnν (t0 )| + (−χ + 2νC1/2 )
Z
t
+ C(χ)
Z
t0
t
2
Z
t
t0
|vnν (s)|2 ds
(|f (s)|2 + |∇zn |2 |zn (s)|2∞ ) + α2 |zn (s)|2 ds
|vnν (s)|2 |∇zn (s)|2∞ .
+
t0
(4.14)
Applying Gronwall lemma, we get
2
|vnν (t)|
Rt
2
≤ |vnν (t0 )| e
t0
(C1 (χ,ν)+|∇zn (s)|2∞ )ds
Rt
+ |∇zn (s)|2 |zn (s)|2∞ )e
s
Z
t
+ C(χ)
t0
(C1 (χ,ν)+|∇zn (σ)|2∞ )dσ
(|f (s)|2 + α2 |zn (s)|2
ds.
(4.15)
where C1 (χ, ν) = −χ + 2νC1/2 .
Inequality (4.15) implies that vnν remains in a bounded set of L∞ (t0 , T ; H), for every T ≥ 0.
Then we go back to (4.14) which shows that vnν remains in a bounded set of L2 (t0 , T ; V ). On
the other hand, we get from the equation and the estimates below proved that ∂v∂tnν is bounded
in L2 (t0 , T ; V 0 ). Hence by a compactness argument, we obtain that
vnν → vν , in L2 ((t0 , T ) × D) strongly,
vnν * vν , in L2 ((t0 , T ; V ) × D) weakly,
so that we can pass to the limit on n (see [14]), and we obtain that vnν verifies the equation
(4.11) in the sence of distributions. Moreover (vnν ) belongs in C([t0 , T ]; H) ∩ L2 (t0 , T ; V ).
Let us denote by ξ = ∇ ∧ u, F = ∇ ∧ f , zr = ∇ ∧ z, g = ∇ ∧ W and set β = ξ − zr , we have
(
dβν + νAβν + B(unu , βν ) + χβν = F dt + αzr − (uν · ∇)zr ,
βν (t0 ) = ∇ ∧ uν (t0 ) − zr (t0 ).
(4.16)
From the incompressibilty condition, we have that
< B(uν , βν ), βν >= 0.
We multiply equation (4.16) by βν and integrate over (t0 , t), and we obtain
|βν (t)|2 + χ
Z
t
t0
|βν (s)|2 ds + 2ν
Z
t
+ C(χ)
t0
Z
t
t0
|∇βν (s)|2 ds ≤ |β(t0 )|2
(|F (s)|2 + α|zr (s)|2 + |uν (s)|2 |∇zr (s)|2∞ ).
Hence, by Gronwall inequality, we have
|βν (t)|2 ≤ |βν (t0 )|2 e−χ(t−t0 )
Z
t
+C(χ)
t0
(|F (s)|2 + α|zr (s)|2 + |unν (s)|2 |∇zr (s)|2∞ )e−χ(t−s) ds.
10
(4.17)
Now let us introduce the elliptic problem

⊥

 4uν = −∇ ξν ,
u |
= 0,
(4.18)
ν ∂D

 ξ |
ν ∂D = 0.
Where ∇⊥ = (D2 , −D1 ). We multiply the first equation by uν and integrate over D. By an
integration by parts and using the estimate (4.13), we have for an arbitrary > 0 (in particular
for = 1/2),
|∇uν (t)|2 ≤ C(|uν (t)|2 + |ξnν (t)|2 ). t ∈ (t0 , T ).
Collecting the estimates (4.15) and (4.17), we have that
2
2
−C1 (χ,ν)t
k vν (t) k ≤k vν (t0 ) k e
+χ
−1
Z
t
t0
k f (s) k2 e−C1 (χ,ν)(t−s) ds.
(4.19)
The estimates (4.19) and (4.14) show that vν remains uniformly bounded in L∞ (t0 , T ; V ) ∩
2
0
ν
L2 (t0 , T ; V ). By the same argument used before, we prove that ∂v
∂t is bounded in L (t0 , T ; V ).
Hence by a compactness argument, we can extract a subsequence (also denoted vν ) such that
vν * v in L2 ((t0 , T ; V ),
and
vν −→ v strongly in L2 ((t0 , T ) × D).
Therefore, we can pass to the limit on n in the equation (4.9) and v satisfies (4.9) in the
distribution sence.
Let us give the definition of global weak solution.
Definition 4.1 We shall say that a stochastic process u(t, ω) is a global weak solution of the
equation (4.1) over the time interval (t0 , T ) if for P a.e. ω
2
u(., ω) ∈ C[t0 , T ; H) ∩ L∞
loc (t0 , T ; V ) ∩ Lloc (t0 , T ; V ),
< u(t) − u(t0 ), φ > +χ
Z
t
=
t0
Z
Z
t
t
< u(s), φ > +
t0
< B(u(s), u(s)), φ >
t0
< f (s), φ > + < W (t) − W (t0 ), φ >,
for all φ ∈ V and for all T ≥ t ≥ t0 .
As a consequence, we give the following theorem
Theorem 4.2 (a) Assume that (4.5) and (4.6) hold. Assume that ut0 ∈ V and f ∈ L2 (t0 , T, V ).
Then, on each interval (t0 , T ) there exists at least a weak global solution for (4.1) with the initial
condition u(t0 ) = ut0 satisfying for P- a.e.ω ∈ Ω
u(., ω) ∈ C[t0 , T ], ; H) ∩ L2 (t0 , T ; V ).
11
Moreover, u is measurable in these topologies and satisfies for P-a.e. ω ∈ Ω and for all t ∈ (t0 , T )
u(., ω) ∈ L∞ (t0 , T ; V ),
Rt
|u(t) − z(t)|2 ≤ |u(t0 ) − z(t0 )|2 e
Z
t
+ C(χ)
t0
t0
(−χ+|∇z|2∞ )
Rt
(|f (σ)|2 + α2 |z|2 + |∇z|2 |z|2∞ )e
σ
(−χ+|∇z|2∞ )
,
(4.20)
|ξ(t) − zr (t)|2 ≤ |ξ(t0 ) − zr (t0 )|2 e−χ(t−t0 )
Z
t
+ C(χ)
t0
and
(|F |2 + α2 |zr |2 + |u|2 |∇zr |2∞ )e−χ(t−σ) dσ.
|∇u(t)|2 ≤ C|ξ(t)|2 + |u(t)|2 .
(4.21)
(4.22)
P
4+γ 2
(b) If in addition for given γ ∈ (0, 1), ∞
σj < ∞, and ξt0 ∈ L∞ (D) and f ∈ L∞ (D)
j=1 λj
then, P-a.e.ω ∈ Ω
ξ(., ω) ∈ L∞ (D × (t0 , T )),
and the solution is unique. Moreover it is progressively measurable in these topologies and it
satisfies for all P-a.s.ω ∈ Ω and for all t ∈ (t0 , T )
|ξ(t) − zr (t)|∞ ≤ |ξ(t0 ) − zr (t0 )|∞ e−χ(t−t0 )
Z
t
+
t0
(|F |∞ + α|zr |∞ + |u|∞ |∇zr |∞ )e−χ(t−σ) dσ,
and
|∇u(t)|∞ ≤ |u(t)|W 1,4 ≤ C |ξ(t)|44 + |u(t)|44
1/4
,
(4.23)
(4.24)
where z is solution of problem (4.8).
Proof. (a) Proved above.
(b) To prove (4.23), we use the maximum principle on the scalar equation (4.16), where the
Galerkin approximation is used. We obtain (4.23) for vν which is also true for v (when ν goes
to 0).
The first part of the estimate (4.24) is given by the continuous embedding W 1,4 (D) ⊂ L∞ (D).
We have to check an estimate for |∇u|4 . Write the equation (4.18) as
−Di2 uj = Dj⊥ ξ.
Here, we use the summation on repetitive indices. Multiply the above equation by |∇u|2 uj , and
integrate over D. By integration by parts, we have that
− < Di2 uj |∇u|2 , uj >= −1/3
Z
D
|∇u|4 + +1/3
12
Z
∂D
k(σ)|u|2 |∇u|2 .
The boundary integral is estimated as
Z
∂D
k(σ)|u|2 |∇u|2 ≤ C| |∇u|2 |H −1/2 (∂D) | |u|2 |H 1/2 (∂D)
≤ | |∇u|2 || |u|2 |H 1
Z
=
D
|∇u|4
1/2 Z
D
|u|4 + 4
Z
D
|u|2 |∇u|2
1/2
,
Using two times the Holder inequality and Young inequality, we obtain that for an arbitrary
> 0, it is less than |u|44 + |∇u|44 . now we turn to the second term of the equation. By the
boundary condition ξ|∂D = 0, and for an arbitrary real number > 0, we get
< Dj⊥ ξ|∇u|2 , uj > =
Z
D
|ξ|2 |∇u|2
≤ |ξ|44 + |∇u|44 .
Collecting all the estimates, we obtain (4.24).
Uniqueness.The proof of uniqueness is given in [2] for χ = 0, but it readily extends to χ 6= 0.
Measurability. When ut0 ∈ V , the processes vnν (t, ω) are progressively measurable in H, by
construction. Analyzing the limiting procedures in the previous steps we deduce that also v(t, ω)
is progressively measurable in H and so u(t, ω) being the difference of two measurable processes
v(t, ω) and z(t, ω). The embedding of V in H being continuous, we can deduce the progressive
measurability of the solution in V . Moreover when ξt0 ∈ L∞ (D), the mollification of the
Navier-Stokes solution is progressively measurable in the space of mollifers (the operation being
continuous), yielding the progressive measurability in the required topology for its pointwise
limit. This completes the proof.
4.3
Existence of stochastic weak attractor
For sake of simplicity, we assume that f ∈ V . We recall that in this section, we are dealing with
the metrics dW , d and δ introduced in section 2, i.e. d is the metric endowed by V , δ and dW
are the ones endowed respectively by H and W. The metric dW is defined by
dW (f, g) = d(f, g) + d∞ (∇ ∧ f, ∇ ∧ g),
where d∞ is the metric of L∞ (D). We denote by |.|W the norm induced by W. We denote by
W = {f ∈ V, such that ∇ ∧ f ∈ L∞ (D)} .
We define the family {S(t, s, ω)}t≥s,ω∈Ω by
S(t, s, ω) :
W
−→ W
us (ω) −→ u(t, s, ω)
Lemma 4.3 The family of mappings {S(t, s, ω)}t≥s,ω∈Ω associated to the Euler equation (4.1)
verify for P.a.e. ω ∈ Ω the evolution property (2.2) on W and the condition (2.1).
13
Proof. The evolution property is obvious, by the uniqueness of the solution in W. Now let us
fix ω ∈ Ω and let us take t ≥ s. Take a sequence uns such that it is P.a.e. ω ∈ Ω dW -bounded
and δ-convergent to us . From the estimates of section 3, we deduce that un (t) is P.a.e. ω ∈ Ω
dW -equibounded and that un ∈ L∞ (0, T ; V ) ∩ W 1,2 (0, T ; H). On the other hand, we have from
the equation (4.1) that P.a.e. ω ∈ Ω
|(un (t) − un (s)) − (W (t) − W (s))|
≤
Z
s
t
|(un (r) · ∇)un (r)|dr +
Z
t
s
| − χun (r) + f (r)|dr +
Z
t
s
|W (r)|dr
≤ (|un |L∞ (0,T ;V ) |un |L∞ (0,T ;H) + |un |L∞ (0,T ;H) + |f | + |W |L∞ (0,T ;H) )|t − s|
≤ C|t − s|.
for all T ≥ t ≥ s ≥ t0 . From Ascoli-Arzelà theorem, there exists a subsequence unk (t) δconvergent to u(t) uniformly on [t0 , T ].
We will prove the following lemma
Lemma 4.4 Let γ ∈ (0, 1) be fixed. For any > 0, there exists α0 > 0 such that for all α ≥ α0
E|z(t)|2(H 2+γ (D))2 < ,
for all t ∈ R.
R
t
Proof. The process z(t) = −∞
e(t−s)(−χ−α) dW (s) solution of (4.8) is an ergodic and stationary
process with continous trajectory in (H 4 (D))2 .
E|z(t)|2(H 2+γ (D))2
= E|A
=
≤
≤
≤
Write
2+γ
2
z(t)|2
2
∞ Z t
X
2+γ
(t−s)(−χ−α
2
E
A
e
σj ej dβj (s)
j=1 −∞
Z
∞ t
2+γ
X
2
(t−s)(−χ−α)
2
E
λ
e
σ
dβ
(s)
j
j
j
j=1 −∞
∞ Z t
X
λ2+γ
e2(t−s)(−χ−α) σj2 ds
j
−∞
j=1
∞
X
1
λ2+γ
σj2 .
2(χ + α) j=1 j
∞
N
∞
X
X
X
1
1
1
2+γ 2
2
λ2+γ
σ
=
λ
σ
+
λ2+γ σj2 ,
j
j
2(χ + α) j=1 j
2(χ + α) j=1 j
2(χ + α) N +1 j
Using the condition (4.7), we can choose α such that the first term of the right hand of the above
equality is less than /2 and N such that the second term is less than /2. This completes the
proof.
14
Lemma 4.5 There exists an absorbing set at time 0 wich is δ-compact.
Proof. By previous lemma, in particular for t = 0, we can choose α such that
χ
−χ + E|z(t)|2H 2+γ ≤ − .
2
By the ergodicity of z we have that
1
−t0
Z
0
t0
|z|2H 2+γ → E|z(0)|2H 2+γ as t0 → −∞.
So for ω ∈ Ω there exists τ (ω) < 0 such that
Z
0
χ
(−χ + |z(r)|2H 2+γ )dr ≤ − (−s), ∀t0 < τ (ω).
4
t0
Moreover, by the continuity of the trajectories of z
Z
sup
0
τ (ω)≤s≤0 t0
(−χ + |z(r)|2H 2+γ )dr ≤ C(ω) < ∞
for some constant C(ω).
Therefore for t = 0, the estimate (4.20) and (4.21) yield
|u(0) − z(0)|2 ≤ |u(t0 ) − z(t0 )|2 eχ(t0 ,ω)+C(ω)
Z
0
+ C(χ)
t0
(|f |2 + α2 |z(r)|2 + |∇z(r)|2 |z(r)|2∞ )eχ(t0 ,ω)+C(ω) dr,
(4.25)
Since |z(s)|2H 3+γ has at most a polynomial growth, when t0 → −∞, the right hand side of (4.25),
(4.21), and (4.23) are almost surely bounded. Collecting all the estimates of theorem 4.2, there
exists a random constant r(ω) such that P.a.e. ω ∈ Ω
|u(0)|2W ≤ r(ω).
We deduce the existence of a dW -bounded and δ-compact random ball which absorbs dW -sets
of W at time 0 from −∞.
As a consequence we give the following theorem
Theorem 4.6 Under conditions of theorem 4.2, there exists a random weak attractor for (4.1)
at time 0, which attracts bounded sets from −∞.
Aknowledgments. Thanks to Franco Flandoli from the University of Pisa for stimulating
conversations and remarks on preliminary versions of this paper.
15
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16